You can find G.I. surplus waterproof bags that are about the same size as the inside of a backpack. Get a new one; well-cared for they will last many years, and aren't terribly expensive.
It is a heavy-duty, rubber-lined nylon bag. Its uses are limited only by the imagination (I use it on cold nights to add a little warmth to the foot of my lightweight sleeping bag). But perhaps the best use I put it to is as a field expedient washing machine.
Take a single day's change of clothes: shirt, trousers, undershirt, drawers, socks and hat, and put them in the bag. Pour a litre of water in, and a tablespoon full of laundry soap powder (or your own bar of soap/squirt of shampoo in a pinch). If you have any REALLY smooth stones nearby, toss one or two small ones in. Squeeze nearly all the air out of the bag, tie it in "gooseneck" form, and pretend you're playing an concertina--squish it back and forth, over and over for about 5 minutes (great chest/tricep workout).
Pour out the water, wring clothes out, pour in a fresh litre of water, and repeat the above to rinse. You probably have to do this twice. Then hang everything, including the bag, to dry.
Not necessary for an overnighter, and there are some who believe that doing one's laundry in the field is vain foppery and contrary to the spirit of camping. I must confess that I'm not among that number. I prefer to be as clean as the situation permits.
Give it a try. Might work for you.